These representations are of fundamental importance in
mathematical physics and homomorphisms from a Lie group
\(G\)  to \(\mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\) , \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\)  or
\(\mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\) 
can be used to obtain semibounded representations of a central
extension by pulling back the representations discussed above.
This construction has been a major source of (projective) representations
for loop groups \(G = C^\infty ({\mathbb {S}}^1,K)\)  and \(\mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)\) 
(cf. [1]}, [2]}, [3]}, [4]}).
